Brownian Motion Effects on the Stabilization of Stochastic Solutions to Fractional Diffusion Equations with Polynomials

Abstract

A class of stochastic fractional diffusion equations with polynomials is considered in this article. This equation is used in numerous applications, such as ecology, bioengineering, biology, and mechanical and chemical engineering. As a result, it is critical to obtain exact solutions to this equation. To obtain these solutions, the tanh-coth method is utilized. Furthermore, we clarify the impact of noise on solution stabilization by simulating our solutions.

Multiplicative Brownian Motion Stabilizes the Exact Stochastic Solutions of the Davey–Stewartson Equations

“…Various approaches for discovering the analytical solutions of PDEs have been presented to overcome this issue. Some examples of the most significant methods are tanh-sech [1][2][3], extended tanh-function [4], the Sine-Gordon expansion [5], the trial function [6], the Darboux transformation [7], the Jacobi elliptic function [8,9], the sine-cosine [10,11], (G /G)-expansion [12,13], Hirota's function [14], exp(−φ(ς))-expansion [15], perturbation [16,17], the qualitative theory of dynamical systems [18][19][20][21], the direct method [22], the Riccati-Bernoulli sub-ODE [23], and the F-expansion method [24].…”

“…Our motivation in this paper is to attain the analytical solutions of the stochastic Davey-Stewartson equations (SDSEs). This work is the first to obtain the analytical solutions of SDSEs (1) and (2). We employ the mapping method to obtain a wide range of stochastic solutions, such as rational, elliptic, trigonometric and hyperbolic functions.…”

Exploring the Exact Solution of the Space-Fractional Stochastic Regularized Long Wave Equation: A Bifurcation Approach

“…where T p is the Jumarie's modified Rieman-Liouville fractional derivative of order p, ρ is the noise strength, H(t) is a white noise (Gaussian process), and QdH is a multiplicative white noise in the Itô sense. Note that the noise-perturbed term is a combination of two terms appearing in [31,32]. Finding solutions for stochastic fractional partial differential equations poses a considerable challenge.…”

Abundant Solitary Wave Solutions for the Boiti–Leon–Manna–Pempinelli Equation with M-Truncated Derivative

“…This has made the development of mathematical techniques for generating accurate solutions to PDEs a substantial and crucial endeavor in the field of nonlinear sciences. Recently, a wide range of approaches, such as (G /G)-expansion [1,2], the mapping method [3], Jacobi elliptic function [4,5], Sardar-subequation method [6], Exp-function method [7], sine-Gordon expansion [8], exp(−φ(ς))-expansion [9], extended trial equation [10], tanh-sech [11,12], F-expansion approach [13], homotopy perturbation technique [14], He's semi-inverse method [15], etc., have been offered as potential solutions to the problem of PDEs.…”